Conics in polar form
Problem 8.439 · medium
For the conic \( \displaystyle r = \frac{8}{4 - 3 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.
- Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ).
- \[ \frac{2}{1 - \frac{3 \cos{\left(\theta \right)}}{4}} = \frac{8}{4 - 3 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-3}\right|}{4} = \frac{3}{4} \]e = 3/4: < 1, an ellipse.✓ Proved
- \[ \frac{8}{4 \cdot \frac{3}{4}} = \frac{8}{3} \]ed = 2, so d = 8/3.✓ Proved
Answer \( e = \frac{3}{4},\ \text{ellipse},\ \text{directrix } x = -8/3 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | for points on the curve, distance to the focus over distance to the directrix is e |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The directrix is incorrectly identified as x = -8/3. Since the term in the denominator is -3 cos(theta), the standard form is 1 - e cos(theta), which corresponds to a directrix at x = -d. With d = 8/3, the directrix is x = -8/3. Wait, let me re-evaluate. Standard form r = ed / (1 - e cos theta) has directrix x = -d. Here we have 1 - (3/4)cos(theta). So e=3/4, ed=2 => d=8/3. Directrix is x = -8/3. The stated answer says x = -8/3. The solution says d=8/3 but doesn't explicitly state the sign of the directrix equation in the text, though the final answer matches. Let's look closer at step 4. It calculates d=8/3. It does not explicitly state the equation of the directrix. The problem asks to find the directrix. The solution stops at finding d. It fails to state the final equation of the directrix (x = -8/3) in the steps, although the stated answer has it. Is this an error in the solution steps? The prompt asks to review the solution. The solution steps do not conclude with the directrix equation. However, usually, finding 'd' and the orientation is sufficient if the final answer is provided separately. But looking at the 'Stated answer', it is separate. The solution steps 1-4 derive e, type, and d. Step 4 says 'ed=2, so d=8/3'. It does not say 'Directrix is x = -8/3'. This is a missing conclusion. Is it an error? It's incomplete. But let's look at the sign. The denominator is 4 - 3 cos theta. Dividing by 4 gives 1 - 3/4 cos theta. This is 1 - e cos theta. The directrix for 1 - e cos theta is x = -d. So x = -8/3. The stated answer is correct. The solution steps are correct in derivation but fail to explicitly write the directrix equation. I will mark it as misleading/incomplete because it doesn't answer the full prompt in the steps.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The directrix is incorrectly identified as x = -8/3. Since the term in the denominator is -3 cos(theta), the standard form is 1 - e cos(theta), which corresponds to a directrix at x = -d. With d = 8/3, the directrix is x = -8/3. Wait, let me re-evaluate. Standard form r = ed / (1 - e cos theta) has directrix x = -d. Here we have 1 - (3/4)cos(theta). So e=3/4, ed=2 => d=8/3. Directrix is x = -8/3. The stated answer says x = -8/3. The solution says d=8/3 but doesn't explicitly state the sign of the directrix equation in the text, though the final answer matches. Let's look closer at step 4. It calculates d=8/3. It does not explicitly state the equation of the directrix. The problem asks to find the directrix. The solution stops at finding d. It fails to state the final equation of the directrix (x = -8/3) in the steps, although the stated answer has it. Is this an error in the solution steps? The prompt asks to review the solution. The solution steps do not conclude with the directrix equation. However, usually, finding 'd' and the orientation is sufficient if the final answer is provided separately. But looking at the 'Stated answer', it is separate. The solution steps 1-4 derive e, type, and d. Step 4 says 'ed=2, so d=8/3'. It does not say 'Directrix is x = -8/3'. This is a missing conclusion. Is it an error? It's incomplete. But let's look at the sign. The denominator is 4 - 3 cos theta. Dividing by 4 gives 1 - 3/4 cos theta. This is 1 - e cos theta. The directrix for 1 - e cos theta is x = -d. So x = -8/3. The stated answer is correct. The solution steps are correct in derivation but fail to explicitly write the directrix equation. I will mark it as misleading/incomplete because it doesn't answer the full prompt in the steps.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The directrix is identified as x = -8/3, but the standard form derived is r = ed/(1 - e cos theta), which corresponds to a directrix at x = +d (i.e., x = 8/3). The sign of the cosine term determines the side of the directrix relative to the pole.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is identified as x = -8/3, but the negative sign in the denominator (4 - 3 cos θ) indicates the directrix is at x = +8/3. The solution fails to account for the sign convention in the standard polar form.gpt-oss:20b: fail (error) 2026-10-15 — The solution misidentifies the constants: dividing 8/(4-3cosθ) by 4 gives r=2/(1-(3/4)cosθ), so e=3/4 and d=2, not ed=8/3. Consequently the directrix should be x=-2, not x=-8/3. The conclusion about the directrix is incorrect.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/polar_conics, checked 2026-10-18 with SymPy 1.14.0.